Class 8 NCERT Solutions - Chapter 12 Exponents and Powers - Exercise 12.2

Exercise 12.2 of Chapter 12 (Exponents and Powers) in the Class 8 NCERT Mathematics textbook builds upon the concepts introduced in Exercise 12.1. This exercise focuses on more advanced applications of exponents, including working with very large and very small numbers, scientific notation, and solving more complex exponential expressions.

Question 1. Express the following numbers in standard form:

(i) 0.0000000000085

(ii) 0.00000000000942

(iii) 6020000000000000

(iv) 0.00000000837

(v) 31860000000

Solution:

(i) 0.0000000000085

= 85/10000000000000 = 85/10¹³ = 85 × 10^-13

= 8.5 × 10 × 10^-13 = 8.5 × 10^1-13 = 8.5 × 10^-12

So, the required standard form is 8.5 × 10^-12

(ii) 0.00000000000942

= 942/100000000000000 = 942/10¹⁴ = 942 × 10^-14

= 9.42 × 10² × 10^-14 = 9.42 × 10^2-14 = 9.42 × 10^-12

So, the required standard form is 9.42 × 10^-12

(iii) 6020000000000000

= 602 × 10000000000000 

= 6.02 × 10² × 10¹³ = 6.02 × 10¹⁵

So, the required standard form is 6.02 × 10¹⁵

(iv) 0.00000000837

= 837/100000000000 = 837/10¹¹ = 837 × 10^-11

= 8.37 × 10² × 10^-11 = 8.37 ×10^2-11 = 8.37 × 10^-9

 So, the required standard form is 8.37 × 10^-9

(v) 31860000000

= 3186 × 10000000

= 3.186 × 10³ × 10⁷ = 3.186 × 10¹⁰

So, the required standard form is 3.186 × 10¹⁰ 

Question 2. Express the following numbers in the usual form

(i) 3.02 × 10^-6

(ii) 4.5 × 10⁴

(iii) 3 × 10^-8 

(iv) 1.0001 × 10⁹

(v) 5.8 × 10¹² 

(vi) 3.61492 × 10⁶ 

Solution:

(i) 3.02 × 10^-6 

= 3.02/10⁶

= 302 × 1/10² × 1/10⁶ = 302/10²⁺⁶

= 302/10⁸ = 0.00000302

So, the usual form is 0.00000302

(ii) 4.5 × 10⁴

= 45/10¹ × 10⁴

= 45 × 10^-1 × 10⁴ = 45 × 10^-1+4

= 45 × 10³ = 45000

So, the usual form is 45000

(iii) 3 × 10^-8

= 3/10⁸

= 3/100000000 = 0.00000003

So, the usual form is 0.00000003

(iv) 1.0001 × 10⁹

= 10001/10000 × 10⁹

= 10001 × 10^-4 × 10⁹ = 10001 × 10^-4+9

= 10001 × 10⁵ = 1000100000

So, the usual form is 1000100000

(v) 5.8 × 10¹²

= 58 X 10^-1 × 10¹²

= 58 × 10^-1+12 = 58 × 10¹¹

= 5800000000000

So, the usual form is 5800000000000

(vi) 3.61492 × 10⁶

= 361492/100000 × 10⁶

= 361492 × 10^-5 ×10⁶ = 361492 × 10^-5+6

= 3614920 

So, the usual form is 3614920

Question 3. Express the number appearing in the following statements in standard form.

(i) 1 micron is equal to 1/1000000 m.
(ii) Charge of an electron is 0.000,000,000,000,000,000,16 coulombs
(iii) Size of bacteria is 0.0000005 m
(iv) Size of a plant cell is 0.00001275 m
(v) Thickness of a thick paper is 0.07 mm.

Solution:

(i) 1 micron = 1/1000000 m

= 1/10⁶ = 10^-6 m

So, the standard from is 10^-6 m

(ii) Charge of an electron = 0.000,000,000,000,000,000,16 C

 = 16/100000000000000000000 = 16/10²⁰ = 1.6 × 10 X10^-20

= 1.6 × 10^1-20 = 1.6 × 10^-19

So, the standard from is 1.6 × 10^-19 C

(iii) Size of Bacteria = 0.0000005 m

= 5/10000000 = 5/10⁷ = 0.5 × 10 × 10^-7

= 0.5 × 10^-6

So, the standard from is 0.5 × 10^-6 m

(iv) Size of plant cell = 0.00001275 m

= 1275/100000000 = 1275/10⁸ = 1.275 × 10³ × 10^-8

= 1.275 × 10^3-8 = 1.275 × 10^-5

So, the standard from is 1.275 × 10^-5 m

(v) Thickness of a Paper = 0.07mm

= 7/100 = 7/10² = 0.7 × 10 × 10^-2

= 0.7 × 10^-1

So, the standard from is 0.7 × 10^-1 mm

Question 4. In a stack, there are 5 books each of thickness 20 mm and 5 paper sheets each of thickness 0.016 mm. What is the total thickness of the stack? 

Solution:

The thickness of Books will be 5 × 20mm = 100mm or 10cm

The thickness of Paper sheets will be 5 × 0.016mm = 0.080mm

Hence, the total thickness is = thickness of books + thickness of paper sheets

= 100mm + 0.080mm = 100.080mm 

or 

1.0008 × 10^-2 mm

Summary

Exercise 12.2 of Chapter 12 on Exponents and Powers delves deeper into the practical applications of exponents, particularly in the realm of scientific notation. This exercise is crucial for students to understand how to work with very large and very small numbers efficiently, a skill that is indispensable in scientific and engineering calculations. The problems in this exercise challenge students to convert between standard form and scientific notation, perform arithmetic operations with numbers in scientific notation, and apply the laws of exponents in more complex scenarios. By mastering these skills, students develop a strong foundation for handling real-world numerical data in fields such as astronomy, physics, and chemistry. The exercise also reinforces the concept of place value and helps students appreciate the power of exponential notation in expressing extreme values concisely. Additionally, working with scientific notation enhances students' ability to estimate and make sense of numerical magnitudes, fostering a deeper understanding of number sense and scale.